Abstract:
Finite-state, discrete-time Markov chains coincide with Markov fields on Z, (which are nearestneighbour Gibbs measures in one dimension). That is, the one-sided Markov property and the two-sided Markov property are equivalent. We discuss to what extent this remains true if we try to weaken the Markov property to the almost Markov property, which is a form of continuity of conditional probabilities. The generalisation of the one-sided Markov measures leads to the so-called ”g-measures” (aka ”chains with complete connection”, ”uniform martingales”, and etc.), whereas the two-sided generalisation leads to the class of Gibbs or DLR measures, as studied in statistical mechanics. It was known before that there exist g-measures which are not Gibbs measures. It is shown here that neither class includes the other. We consider this issue in particular on the example of long-range, Dyson model, Gibbs measures. The proof is based on the phenomenon of ”entropic repulsion” in these models. (Work with R.Bissacot, E.Endo and A. Le Ny)
Biography:
Aernout van Enter is professor emeritus at Groningen University, the Netherlands, where he also obtained his education. He has worked at the University of Heidelberg, Germany, the Technion at Haifa, Israel, and the University of Texas at Austin, USA. His research interest is in mathematical statistical physics. He has worked on the theory of Gibbsian and non-Gibbsian measures, on various types of phase transitions in lattice models, on disordered systems and spin glasses, on the theory of aperiodic order and on bootstrap percolation.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai